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June 15, 2016

Section 5: Expected Value vs. Expected Growth (EV vs. EG)

The difference between Expected Value and Expected Growth is important to bettors looking to maximize their long term edge, a topic I’ve already flirted with in my KC articles. “Expected Value” is simply the average result we would expect from an event if that event were repeated an infinite number of times. It can be positive or negative; if a bettor places a $100 wager on a game at -110 odds with a 57% win probability, he may end up with $0 or $190.91, but his EV is $108.82 (+$8.82), making the bet +EV. Conversely, placing a $100 bet on a Roulette game, be it on a specific number, set of numbers, black, red, evens or odds (Roulette wheels have 18 black, 18 red and 2 green numbers, and pay out 36-1 on a single number, 2-1 on black/red/evens/odds, etc.) has an expected return of $94.74, or -$5.26, making the bet -EV.

Generally, you want to take bets that are +EV and avoid bets that are -EV, and you want to size your bets with some sort of Kelly based logic, approximately setting your bankroll for optimal growth. Most people assume that’s where the discussion ends, but in fact bankroll growth is more complex than it appears at first glance. (Are you sensing a trend here?) The reason for this is that winning bettors must always consider their opportunity cost – when a bettor loses a bet, they lose not only the amount they wagered, but also the future expectation of earnings from that money.

Thus, they prefer events with shorter odds to events with longer odds. I have already explained this indirectly (remember that optimal KC wagering is edge over odds) but it should also make sense intuitively. If the gambling millionaire from the first article offered you a 10% shot at $50 million or a 95% shot at $5 million, which would you take? Unless you’re already a multi millionaire, the obvious answer is that you would take the 95% shot at the $5m, even though it’s -EV compared with 10% at 50m. We can rationalize this by citing long term expected bankroll growth.

Consider a more practical example: Let’s say North Carolina has been given an 28% chance to win an NCAA Basketball National Title (paying out 3.57 to 1), but you believe that their true chance is 35%. Marquette has been given a 2% chance to win (paying out 50 to 1), but you believe that their true chance is 3%. Which of these is a more attractive bet?

The EV of betting $100 on UNC is $124.95, or +$24.95. (35% of the time you get $357, 65% of the time, you get $0).

The EV of betting $100 on Marquette is $150, or +$50 (3% of the time you get $5,000, 97% of the time you get $0).

So, betting on Marquette is much better, right? Betting on Marquette is more than twice as +EV as betting on UNC. Marquette will win the title 50% more often than oddsmakers expect them to, while UNC will win only 25% more often than they’re expected to. However, we of course have to decide how much of our bankroll we are going to bet on this advantageous position. If we do a Kelly calculation, we find that we should optimally wager (50*.03-.97)/50 = 1.06% of our bankroll on Marquette. We should optimally wager (3.57*.35-.65)/3.57 = 16.79% of our bankroll on UNC.

A bet on Marquette is +50% EV, and we would put 1.06% of our bankroll on them, so by betting on the Golden Eagles, we would expect to grow our bankroll by 0.53%.

A bet on UNC is +24.95% EV, and we would put 16.79% of our bankroll on them, so by betting on the Tarheels, we would expect to grow our bankroll by 4.19%

Thus, betting on UNC is massively (nearly 8 times) more +EG than betting on Marquette, despite the fact that Marquette has a higher +EV. Since almost all of my picks are 53-60% winners at roughly -110 odds, we wouldn’t use EG to compare my picks on one game to my picks on another game. We can, however find this useful when examining my results as compared with other potential non-sports investments. Over the past 22 years, my year to year performance (across all sports in a given year) has been remarkably consistent and has been both massively +EV as well as +EG. When we discuss the Sharpe Ratio, we will examine what percentage of your investments you should invest into my picks.